# Fundamental Theorem of Calculus (FTC)

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 4.9 – Antiderivatives

• Section 5.3 – The Fundamental Theorem of Calculus

• Section 5.4 – Indefinite Integrals and the Net Change Theorem

### Expected Educational Results

• Objective 01–01: I can state the Fundamental Theorem of Calculus.

• Objective 01–02: I can explain the meaning of the Fundamental Theorem of CalculusPart I.

• Objective 01–03: I can explain the meaning of the Fundamental Theorem of CalculusPart II.

• Objective 01–04: I can determine the properties of an area function using FTC-I and Calculus I.

• Objective 01–05: I can evaluate indefinite integrals using the Fundamental Theorem of CalculusPart I.

• Objective 01–06: I can evaluate definite integrals using the Fundamental Theorem of CalculusPart II.

• Objective 01–07: I can use the Net Change Theorem to identify the meaning of ${\int }_{a}^{b}{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)\phantom{\rule{0.167em}{0ex}}\text{d}x$$\int_a^b{f^{\,\prime}(x)\,\text{d}x}$.

• Objective 01–08: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.

• Objective 01–09: Given acceleration of an object in motion along a line, I can find the object’s velocity, displacement, and the total distance traveled.

### Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

## FTC-II

#### Theorem: Fundamental Theorem of Calculus - Part II [FTC-II]

Let $f\left(t\right)$$f(t)$ be continuous on $\left[a,b\right]$$[a,b]$. Let $F\left(t\right)$$F(t)$ be any antiderivative of $f\left(t\right)$$f(t)$. Then ${\int }_{a}^{b}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt=F\left(b\right)-F\left(a\right)$$\displaystyle \int_a^b{f(t)\,dt}=F(b)-F(a)$.

### Check Conditions of FTC-II

1. $f\left(t\right)$$f(t)$ is the black graph. Is $f\left(t\right)$$f(t)$ continuous on $\left[1,12\right]$$[1,12]$? Explain. NOTE: A function is continuous on an interval IF:

• there are NO holes in the graph;

• there are NO jump discontinuities in the graph; and

• there are NO infinite discontinuities in the graph.

#### Activity 16

NOTE: This DESMOS activity will help you understand the Fundamental Theorem of Calculus - Part II (FTC-II). For a good proof (which is beyond the expectation for this course) of both FTC-I and FTC-II, see https://math.berkeley.edu/~peyam/Math1AFa10/Proof of the FTC.pdf.

Procedure:

Let $g\left(x\right)={\int }_{1}^{a}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$g(x)=\int_1^a{f(t)\,dt}$. In the DESMOS page,

1. Move the slider for the parameter $a$$a$ [Box 3] to a value of $a=12$$a = 12$.

2. Identify the net area [the value listed in Box 2] under $f\left(t\right)$$f(t)$ [black curve].

3. From the graph, identify the value of the antiderivative $F\left(x\right)$$F(x)$ function [green curve] when $a=12$$a=12$.

4. From the graph, identify the value of the antiderivative $F\left(x\right)$$F(x)$ function [green curve] when $a=1$$a=1$.

5. Evaluate $F\left(12\right)-F\left(1\right)$$F(12)-F(1)$ and compare to the net area under $f\left(t\right)$$f(t)$ [black curve]. What do you notice?